In this talk, we explore the relative complexity of different numerical bases, such as d-ary expansions and continued fractions, focusing on cases with zero entropy ( « predictable » expansions). For two expansions S1 and S2 with positive entropies h(S1) and h(S2), Dajani and Fieldsteel (2001) showed each symbol of S2 costs, on average, h(S1)/h(S2) symbols from S1.
We extend this result to sources with zero entropy by introducing a renormalized entropy function. When such a function exists (write f1 for S1 and f2 for S2), we show that the symbol generation rate satisfies lim f2(m)/f1(n)=1. For positive entropy, f is linear, but for the zero-entropy case, it is sub-linear. We give natural examples of this function, in particular from discrete codings of lines (Sturmian words).
Based on joint work with Valérie Berthé, Eda Cesaratto and Martín D. Safe.
Localisation
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